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no relations are to be introduced between a2, /32, and a1} {j^ Since the coefficient of c2 in (13) is constant, if we consider as variable such quantities only as may change in passing from the one group to the other, it will be easily seen that the intensity of either component depends on the same four constants A, 5, 1 C, D as before; and since these have the same values for equivalent groups, it follows that in the most general mode of resolution the components of any two such groups have the same intensities respectively.
15. THEOREM. If two equivalent groups be each resolved in the same manner into two oppositely polarized streams, and these be recombined after their vibrations have been diminished in two different given ratios, the resultant groups will be equivalent.
Let each of the resultant groups be resolved into two oppositely polarized streams 0, E, and let 0', E' denote the streams which furnished 0, E respectively. It is easily seen that the streams 0', Er are both polarized, though not in general oppositely. Were there any occasion to determine the nature of the polarizations, it might easily be done by following a process the reverse of that by which the modified are deduced from the original groups. Thus, if it were required to determine the polarization of 0', we should resolve 0 into streams oppositely polarized in the manner originally given, augment the vibrations in ratios the inverse of those in which they are actually diminished, and recornbinc the streams so obtained. For our present purpose, however, it is sufficient to observe that the polarizations of 0', E' depend only on the mode of resolution, and not on the nature of the original group, and that therefore they are the same for the one group as for the other. The intensity of (/ is not the same as that of 0, as in the case considered in Art. 13, but bears to it a ratio depending only on the mode of resolution, and therefore the same for each of the two original groups. The same is true of the intensity of E' compared with that of E. Now by Art. 14 the intensities of 0', E' arc the same for the two original groups, and the intensities of 0, E bear to those of 0', E' respectively, ratios the same for the two groups: therefore the intensities of 0, E are the same for the two final groups; but 0, E are any two oppositely polarized cora- ponents of these groups ; therefore these groups are equivalent.